The question I was asked is this: is there a difference between praefecti equitum, praefecti equitum alae, and praefecti alae?

A quick rider, the term praefectus equitum is used of individuals in the Republican and Triumviral periods to refer to men who commanded more than a single ala of cavalry. This question doesn't relate to them.

My suspicion is that that there was no difference between the posts, and that what we have on inscriptions is merely a different way of recording them. That is to say that an inscription that names a post as praefectus alae Gallorum I means "praefectus equitum in command of the first ala of Gauls", and that the "equitum" has been dropped because the name of the unit makes it clear that the post was that of praefectus equitum. The formula "praefectus equitum alae X" suggests that whoever wrote the inscription was somewhat pedantic: once praefectus alae came into common use, there was no need to specify that the commander of a cavalry ala was a praefectus equitum (cf. also the unusual phraseology in CIL 6.41050 (Roma)). During the first century AD, the cursus inscriptions of officers increasingly name the units they commanded, and this would only have accelerated the process. The phenomenon can also, perhaps, be seen in the occasional dropping of the word "militum" from tribunus militum when the legion in which that individual served is identified ("tribunus legionis II Augusta" etc.).

So, there's the question, and my views so far. However, I'm not that strong on the terminology of the post-Claudian period, and I wondered if anyone had any thoughts?

The short answer is, in my opinion, yes -- the two posts are essentially the same.

My own theory, which I briefly wrote up for Jasper's Ancient Warfare magazine, is that the early praefecti equitum use that terminology because the first generation of cavalry units had no names. If an equestrian wished to record his military service, he could only specify that he had served as "prefect of horse".

I believe that these early units were identifiable only by their commander's name, so we get examples like ala Scaevae ("Scaeva's squadron"). In theory, when their commander moved on to his next posting, the regiment could take its new commander's name. Or it could nostalgically hark back to a previous commander (it's first ever?), so we get examples like ala Petriana ("the Petrian regiment", meaning "Petra's old regiment").

Once these names start to stick, future commanders can record their military service as prefect of a particular named ala. First as praef. equit. alae whatever, and gradually (as you said) dropping the equitum, which was now redundant. At the same time, with the increase in recruitment, we start to get a few regiments named after the nationality of their original recruits (e.g. ala Hispanorum) -- but it's remarkable how few there are (by contrast with the cohorts).